The major point of the excerpt was that some events like finding people with matching birthdays can be a rare event or a common event depending on what you mean by a matching event. In a room of 23 people there is about a 6% chance one of them will share a birthday with you, but a greater than 50% chance that two of the 23 people in the room will share a birthday (see a description of the birthday problem here). An intuitive explanation for this is when you’re looking for someone who shares your birthday you have limited the possibilities – they must match your birthday – but when we allow anyone to match any other person, the possible number of combinations of people matching birthdays goes up a lot. So its a lot more likely. The maths for this is at the above link. I don’t need to repeat it here. The point is, a subtle change in the question can have big consequences for the answer.

The excerpt then goes on to talk about the Bulgarian Lottery coincident that occurred in 2009. In one draw, the winning numbers were 4, 15, 23, 24, 35, 42. “Amazingly” the next draw, the same numbers again. There was outrage with people calling the draw rigged and there was even a full government investigation. As an aside, if you could rig the draw why would you repeat the numbers? Seems silly to do that. Why not just rig the draw with some other numbers? Anyway…. the question is, how surprised should we be that this happens at all?

The Bulgarian lottery is a 6 number draw from 49 numbers. So the probability of getting the right numbers is 1/(49!/(6!*43!)) or one in 13,983,816. Once you have a particular set of numbers the chance that any other specific draw will have the same numbers is one in 13,983,816 – its the same as picking the numbers in the first place. This seems like a pretty unlikely event. But this is like finding someone who has the same birthday as you. You are trying to match one specific draw to another. What happens when you have many draws and you are trying to find a match between any of those two draws? The Bulgarian lottery has been going on for a while, so there have been many draws and so the chances that a match would happen among all these draws is much less amazing. For example, among 50 draws there are 1,225 ways we could match 2 of the draws. Quickly, the number of ways to find a match goes up and as a result the probability of a match happening goes up as well.

Now some skeptics might say, “But this was two draws in a row!!! Not a draw this week matching a draw from 5 years ago. This is very special because of the serial way in which it happened!”. I felt this objection was glossed over in the SciAm excerpt. Thinking that the consecutive nature of the draws is special in this case is thinking about it all wrong. Actually its thinking yourself into the special circumstance. You are saying that the two draws in a row is a special thing. But this is the same as finding that one person who matches your birthday.

Another way to think about this sort of thing is to consider the following. Instead of drawing 4, 15, 23, 24, 35 and 42, imagine the lottery drew 1, 2, 3, 4, 5 and 6. Would you think this was amazing? Rigged? Would it be all over the internet and in all the papers? Yes it would. But is it any more unlikely than 4, 15, 23, 24, 35 and 42? No, its not. They have the same chance of happening… one in 13,983,816. The only reason it would be considered special is because the numbers are consecutive. By why is that any more special than starting at 4, adding 11, adding 8, adding 1, adding 11 again and then adding 7? Its only because of the value placed on consecutive numbers by our minds. We humans would say consecutive draws has more meaning or is more special than one draw matching a draw from 5, 10 or 15 years ago. In terms of random events neither is more special than the other.

Your perspective influences your expectation, but not reality. Uncommon things will still happen all the time, just not the sort of uncommon things you might expect to happen.

obviously doesn’t converge. Right? This sum doesn’t actually evaluate to a number, does it? Well I wouldn’t think so – no part of my intuition suggests this series is limited. So imagine how surprised I was (and perhaps you too) when I learned this Maths conundrum has been spreading around the internet early this year due to a few videos from the Numberphile guys. Their answer?

What? Really? ? There is something odd here, clearly. How can this be true? This is even beyond breaking intuition, this is just plain wrong. So why is this even taken seriously? The ‘proof’ comes from a field of Maths called Mathematical Analysis and infinite series. I wont go over the proof, there are examples here, here and here. However, they all depend on the following equality:

This is true for a Cesàro summation. That is, its true because of a definition of how to sum a series that doesn’t actually converge to a value. This is fairly well accepted and I suppose if you define the process of Cesàro summation to be a method of summing non-converging sequences then that is just fine… by definition. For me, however, this doesn’t sit right. This is not a sum that comes from the axioms that lead to arithmetic, the so called Peano axioms (I don’t have problems with the Peano axioms – I doubt you do/would too). But I do have a problem with Cesàro summation as a definition. Since Cesàro summations have results like 1-1+1-1… = 0.5 that contradict other forms of the sum like…

and

the only way the sum could be evaluated to 0.5 is by a specific definition that excludes the above sums. But why should I select this definition? Do I accept it as an axiom? If I do, how do I account for the fact that this axiom has two other intuitive sums? Obviously it is a bad axiom and therefore a bad definition. I really don’t understand why anyone would accept the Cesàro summation as anything useful at all. It immediately leads to inconsistencies. Its useless.